GATE | GATE-CS-2006 | Question 18

We are given a set X = {x1, …. xn} where xi = 2ⁱ. A sample S ⊆ X is drawn by selecting each xi independently with probability pi = 1/2. The expected value of the smallest number in sample S is:
(A) 1/n
(B) 2
(C) sqrt(n)
(D) n

Answer: (D)
Explanation: E = (1/(2^1))*(2^1) + (1/(2^2))*(2^2) + … (1/(2^n))*(2^n)
= 1+1+…1 (n times addition of 1)
= n
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